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.  ^-THE.  VENTILATION  OF  COAL-MINES.    By  W 


TRANSITION  CURVES. 


A  FIELD  BOOK  FOR  ENGINEERS, 


CONTAINING 


RULES  AND  TABLES  FOR  LAYING 
OUT  TRANSITION  CUEVES. 


BY 


WALTER   G.   FOX, 

CIVIL  ENGINEEK. 


NEW  YORK: 

D.  VAN  NOSTRAND    COMPANY,  PUBLISHERS, 
23  MURRAY  AND  27  WARREN  STREETS. 

1893. 


COPYRIGHT,  1803. 
Do  VAN  NOSTRAND  COMPANY. 


PREFACE. 

In  this  work  the  endeavor  has  been  to 
condense  the  essential  facts  and  principles 
constituting  the  theory  of  the  transition 
curve  and  to  demonstrate  how  the  curve 
can  be  conveniently  laid  out  in  field  prac- 
tice. 

The  same  methotl  nas'been  adopted  as  is 
usually  employed  in  running  circular  curves 
so  that  the  field  operations  may  be  more 
readily  comprehended  and  performed. 

WALTER  G.  Fox. 

NEW  YOKK,  October  3d,  1893. 


CHAPTER    I. 


INTRODUCTION. 

Near  the  ends  of  a  railway  curve  the 
curvature  should  be  gradually  diminished 
to  enable  trains  at  high  speed  to  deviate 
gradually  from  the  tangent  and  avoid  any 
shock  caused  by  changing  too  suddenly 
from  a  straight  line  to  a  sharp  curve. 

A  transition  curve  is  generally  used  to 
connect  the  circular  curve  with  the  tangent 
as  it  makes  easy  transition  between  them. 

Moreover,  as  the  outer  rail  of  a  curved 
track  is  elevated  above  the  inner  rail,  a 
transition  curve  will  unite  better  with  the 
tangent ;  for  the  super  elevation  of  the 
outer  rail  is  proportional  to  the  degree  of 
curve  which  is  least  at  the  initial  point. 


CHAPTER  II. 

A  transition  curve  is  a  compound  curve 
with  many  changes  of  radius.  It  has  nearly 
the  form  of  the  cubic  parabola. 

The  curve  is  laid  out  on  chords  with  the 
transit  in  the  same  manner  as  circular 
curves.  A  uniform  chord  length  of  10 
feet  has  been  arbitrarily  assumed  to  facili- 
tate calculation,  and  the  degree  of  curve 
changed  at  the  end  of  every  chord.  The 
deflection  angles  have  been  computed  and 
will  be  found  in  the  tables. 

To  compute  the  deflection  angles  it  was 
necessary  to  have  the  co-ordinates  of  every 
point  of  compound  curve.  The  sines  and 
cosines  of  the  different  angles  between  the 
tangent  and  the  chords  of  the  curves  mul- 
tiplied by  10  will  give  the  required  co-or- 
dinates. 

In  Fig.  I.  T  is  the  beginning  of  the 
curve.  The  distance  measured  along  the 
tangent  in  the  direction  of  L  is  the  latitude 
and  in  the  direction  of  D  at  a  right  angle 


ITig.  1.. 


with  the  tangent  is  the  departure  of  the 
various  points  of  compound  curves. 

For  example  let  us  refer  to  Table  L, 
which  begins  with  a  0°  10'  curve  and 
compounds  into  a  sharper  curve  at  the  end 
of  each  chord,  the  degree  of  curve  chang- 
ing to  0°  20',  0°  30',  0°  40',  etc.  See  Fig.  2. 


1 
ft 

Degree  of 
curve. 

Angle   of 
chord. 

Cosine 
xlO. 

Sine 
xlO. 

Co-ordinates 

Lati- 
tude. 

Depar- 
ture. 

10 

0°   10' 

0°     O7  30" 

10.000000 

0.001454 

10.000000 

0.001454 

20 

0°   20' 

0°     2'  00" 

9.999998 

0.005818 

19.999998 

0.007272 

30 

0°   30' 

0°     4'  30" 

9.999991 

0.013090 

29.999989 

0.020362 

40 

0°   40' 

0°     8'  00" 

9.999973 

0.023271 

39.999962 

0.043633 

50 

0°   50' 

0°  12'  30" 

9.999933 

0.036361 

49.999895 

0.079994 

Having  calculated  the  co-ordinates  of 
the  different  points  the  deflection  angles 
were  determined  by 

-  =  tan.  T. 


Example:   To  find  the  deflection  angle 
for  the  first  point  in  Table  I. 


C.  0°  20' 


C.  0   10 


10 

Dep.,   0.001454      log.       —3.1625644 
Lat.,  10.000000      log.  1.0000000 

Ans.     Del  angle,  0°  0'  30"     log.  tan.   6.1625644 

The  difference  in  the  degree  of  curve 
between  the  different  arcs  of  the  transition 
curve  should  be  maintained  as  nearly  as 
possible  between  the  terminating  arc  and 
the  circular  curve. 

In  the  field  the  transit  should  be  placed 
at  T  and  the  angles  deflected  from  the 
tangent.  In  case  there  should  be  an  ob- 
stacle making  a  point  invisible  at  T,  the 
next  point  should  be  measured  on  the  lung- 
chord.  The  intermediate  point  can  then 
be  easily  set. 

To  find  the  length  of  the  long  chord  to 
any  point  on  a  curve.  Fig.  3. 

Let  C  be  the  length  of  chord,  L  the 
latitude  of  the  point,  and  A  the  deflection 
angle. 

c=  L 

cos  A 
Example : 

To  find  the  length  of  the  long  chord  to 
the  8th  point  of  the  curve  in  Table  III. 


11 


12 

Lat.  79.991  log.          1.9030411 

Def.  angle,      0°  38'  15"  log.  cos.  9.9999731 

Ans.     Long  chord,   79.996  log.          1.9030680 


CHAPTER  III. 


EXPLANATION  OF  TABLES. 

The  curvature  increases  more  rapidly  in 
each  succeeding  table. 

Table  I.  begins  with  a  0°  10'  curve  which 
is  the  difference  in  curvature  between  the 
first  arc  of  each  table,  the  degrees  of  curve 
being  0°  10',  0°  20',  0°  30',  etc.  The  de- 
gree of  curve  of  the  first  arc  is  equal  to- 
the  difference  between  all  the  other  arcs  in 
the  same  table,  so  that  the  sixth  arc  in 
Table  I.  is  a  1°  curve  and  the  sixth  arc  in 
Table  VI.  is  a  6°  curve. 

Column  I.  gives  the  length  of  curve  ac- 
cording to  the  central  angle  consumed 
which  will  be  found  on  the  same  line  in  a 
different  column,  and  also  the  other  corres- 
ponding parts  of  the  tables  under  their 
respective  headings. 

Column  II.  has  the  degree  of  curve  of 
each  separate  arc. 


14 


Column  III.  contains  the  deflection  an- 
gles. It  will  be  observed  that  the  value 
of  the  angles  has  been  calculated  to  the 
nearest  second.  Such  accuracy  is  not  ex- 
pected in  practice  but  was  necessary  in  or- 
der to  correctly  determine  the  long  chords. 

Column  IV.  gives  the  central  angles  of 
the  curves. 

Columns  V.  and  VI.  have  the  co-ordi- 
nates of  every  point  of  compound  curve, 
By  using  the  co-ordinates  in  place  of  the 
deflection  angles  the  curve  may  be  laid  out 
with  offsets  from  the  tangent. 

Column  VII.  has  the  long  chords  to  ev- 
ery point  on  the  curves. 

Column  VIII.  contains  numbers  by  which 
the  tangents  of  half  the  intersection  angles 
of  the  curves  should  be  multiplied  and  the 
product  added  to  the  tangent  distances  as 
a  correction. 


CHAPTER  IV. 


PROBLEMS. 

Given,  a  5  degree  curve,  to  run  a  transi- 
tion curve  connecting  it  with  a  tangent 
having  an  intersection  angle  of  20  degrees, 

If  we  select  the  curve  from  Table  VL 
its  length  will  be  40  feet,  consisting  of  4 
arcs,  and  the  central  angle  will  be  1  de- 
gree. As  this  must  be  repeated  at  the 
other  end  of  the  curve,  the  central  angle 
should  be  multiplied  by  2  and  the  sum  sub- 
tracted from  the  intersection  angle.  The 
remainder  will  be  the  central  angle  of  the 
circular  curve.  To  find  its  length  divide 
the  central  angle  by  the  degree  of  curve. 

Example : 

Intersection  angle,  20° 

Central  angle  of  trans,  curve  X  2,        2° 

5/  18° 


Length  of  circular  curve,  360  ft. 


16 

To  find  the  tangent  distance  V. 

Let  M.  be  the  length  of  circular  curve 
that  the  central  angle  of  transition  curve 
consumes,  a  d  the  apex  distance  of  cir- 
cular curve  and  L  the  latitude  of  the  ter- 
minal point  of  the  transition  curve. 


—  M, 

Example  : 

Apex.  dist.  of  5°  curve  for  20°,         202  .  120 
Lat.  of  ter'l  point  of  trans,  curve,       39  .  998 

242.118 
Length  of  5°  curve  to  consume  1°,     20.000 

Ans.  V  =  Tan.  distance,  222.118 

A  correction  should  be  added  to  the  pro- 
duct which  will  not  materially  change  the 
result  unless  the  length  of  curve  exceeds 
70  feet,  or  the  intersection  angle  contains 
more  than  50  degrees. 

Rule:  Multiply  the  tangent  of  half  the 
intersection  angle  by  the  distance  in  column 
8  corresponding  to  the  terminal  point  of 
the  curve.  ' 


OF  THE 

UNIVERSITY 
17 


Example  : 

1/2  intersection  angle—  10° 

Tan.  =  .1763  X  .086  ~  .015 

then    Y    =  -  222.118 

Correction  .015 

Tan.  dist.  -  222.133 

With  curves  of  short  radius  there  is  a 
slight  error  in  the  preceding  formula,  as  M 
does  not  always  have  exactly  its  true  value. 
The  error,  however,  is  too  small  to  take 
into  account  in  field  practice.  In  the  above 
example  it  is  only  one-thousandth  of  a 
foot.  In  a  ten  degree  curve  the  error  is 
about  three-tenths. 

The  true  value  of  M  can  be  determined 
by  multiplying  the  cosine  of  half  the  cen- 
tral angle  of  transition  curve  by  the  length 
of  circular  curve  it  has  consumed. 
Example  : 

1/2  central  angle  of  trans. 

curve  =  0°  30'  log.   cos.    9.9999835 
Length  of  circular  curve  con- 

sumed =  20  feet        log.   1.3010300 

Ans.  M,  =  19.999  ft.          loff.   1.3010135 


18 


Having  measured  the  tangent  distances, 
the  transition  curves  should  be  first  laid 
out  and  then  connected  by  running  the  cir- 
cular curve.  See  Fig.  4. 


Fig. 


If  there  are  two  circular  curves  of  dif- 
ferent radii,  their  ends  will  be  a  short  dis- 
tance apart,  the  sharper  curve  being  on  the 
inner  side,  which  can  then  be  eased  off  to 
unite  with  the  other  curve.  See  Fig.  5. 

On  railway  location  it  may  be  desirable 
to  first  lay  out  the  circular  curve,  neglect- 
ing the  transition  curves  until  the  time  of 
construction. 


19 


In  this  case  theN  two  terminal  points  of 
the  transition  curves  should  be  located  with 
their  co-ordinates,  the  transit  set  over 
either  point  and  a  foresight  taken  on  the 
other  and  the  curve  run  in. 

Compound  curves  should  not  be  located 
in  this  manner,  as  a  backsight  is  then  re- 
quired. 


TRANSITION  CURVE  TABLES. 


23 
TABLE   I. 

1  Length  of  1 
1  Curve. 

j  Degree  of 
1  Curve. 

I       "§> 

1.1. 

ii 
w 

Co-ordinates 

Long 
Chord. 

|  Corr'n  for 
f  Tang.Dist. 

Lati- 
tude. 

Depar 
ture. 

10 

0°  10' 

0°     0'  30" 

0°    1'   0" 

10.000 

0.001 

10.000 

.000 

20 

0°  20' 

0°     I'  11" 

0°    3'   0" 

20.000 

0.007 

20.000 

.004 

30 

0°  30' 

0°     2'  17" 

0°    6'   0" 

30.000 

0.020 

30.000 

.007 

40 

0°  40' 

0°     3'  41" 

0°  10'   0" 

40.000 

0.043 

40.000 

.014 

50 

0°  50' 

0°     5'  30" 

0°  15'   0" 

50.000 

0.080 

50.000 

.029 

60 

1°  00' 

0°     V  33" 

0°  21'   0" 

60.000 

0.132 

60.000 

.045 

70 

1°  10' 

0°     9'  58" 

0°  28'   0" 

69.999 

0.203 

69.999 

.061 

80 

1°  20' 

0°  12'  43" 

0°  36'   0" 

79.999 

0.296 

79.999 

.086 

90 

1°  30' 

0°  15'  49" 

0°  45'   0" 

89.998 

0.414 

89.999 

.126 

100 

1°  40' 

Oc  19'  15" 

0°  55'   0" 

99.997 

0.560 

99.998 

.168 

110 

1°  50' 

0°  23'    0" 

1°    6'  0" 

109.995 

0.736 

109.997 

.208 

120 

2°  00' 

0°  37'    4" 

1°  18'  0" 

119.993 

0.945 

119.996 

.265 

130 

2°  10' 

0°  31'  30" 

1°  31'  0" 

129.990 

1.191 

129.995 

.341 

140 

2°  20' 

0°  36'  15" 

1°45'  0" 

139.986 

1.476 

139.994 

.417 

150 

2°  30' 

0°  41'  19" 

2°    0'  0" 

149.981 

1.803 

149.992 

.494 

160 

2°  40' 

0°  46'  44" 

2°  16'  0" 

159.974 

2.175 

159.989 

.593 

170 

2°  50' 

0°  52'  30" 

2°  33'  0" 

169.965 

2.596 

169.985 

.717 

180 

3°  00' 

0°  58'  35" 

2°  51'  0" 

179.954 

3.067 

179.980 

.843 

190 

3°  10' 

1°    4'  59" 

3°  10'  0" 

189.940 

3.591 

189.974 

.967 

200 

3°  20' 

1°  11'  45" 

3°  30'   0" 

199.923 

4.173 

199.  966 

1.119 

210 

3°  30' 

1°  18'  50" 

3°  51'  0" 

209.903 

4.814 

209.958 

1.302 

220 

3°  40' 

1°  26'  14", 

4°  13'  0" 

219.878 

6.517 

219.947 

1.487 

230 

3°  50' 

1°  34'    0" 

4°  36'  0" 

229.848 

6.286 

229.934 

1.6T1 

240 

4°  00' 

1°  42'    5" 

5°    0'  0" 

239.814 

7.123 

239.920 

1.889 

24 

TABLE  II. 

Length  of  1 
Curve.  | 

!«| 

g>3 
ft 

Deflection 
Angle. 

Central 

Angle. 

Co-ordinates 

Long 
Chord. 

Corr'tJ  for 
Tang.Dist.  ( 

Lati- 
tude. 

Depar 
ture. 

10 

0°  20' 

o°   i'  o" 

0°  2'  0" 

10.000 

0.003 

10.000 

.002 

20 

0°  40' 

0°     2'  24" 

0°  6'  0" 

20.000 

0.014 

20.000 

.005 

30 

1°  00' 

0°    4'  35" 

0°12'  0" 

30.000 

0.040 

30.000 

.014 

40 

1°  20' 

0°     7'  28" 

0°20'  0" 

40.000 

0.087 

40.000 

.029 

50 

1°  40' 

0°  11'    0" 

0°30'  0" 

50.000 

0.160 

50.000 

.051 

60 

2°  00' 

0°  15'    7" 

0°42'  0" 

59.999 

0.264 

59.999 

.081 

70 

2°  20' 

0°  19'  59" 

0°56'  0" 

69.998 

0.407 

69.999 

.122 

80 

2°  40' 

0°  25'  29" 

1°12'  0" 

79.996 

0.593 

79.998 

.174 

90 

3°  00' 

0°  31'  40" 

1°30'  0" 

89.993 

0.829 

89.996 

.240 

100 

3°  20' 

0°  38'  30" 

1°50'  0" 

99.989 

1.120 

99.995 

.320 

110 

3°  40' 

0°  45'  58" 

2°12'  0" 

109.983 

1.471 

109.993 

.415 

120 

4°  00' 

0°  54'    9" 

2°36'  0" 

119.974 

1.890 

119.989 

.529 

130 

4°  20' 

1°     3'     0" 

3°02'  0" 

129.962 

2.382 

129.983 

.662 

140 

4°  40' 

1°  12'  29" 

3°30'  0" 

139.946 

2.951 

139.977 

.813 

150 

5°  00' 

1°  22'  39" 

4°  0'  0" 

149.924 

3.605 

149.967 

.988 

160 

5°  20' 

1°  33'  29" 

4°32'  0" 

159.897 

4.349 

159.956 

1.185 

170 

5°  40' 

1°  44'  59" 

5°  6'  0" 

169.861 

5.189 

169.940 

1.407 

180 

6°  00' 

1°  57'    9" 

5°42'  0" 

179.817 

6.130 

179.921 

1.655 

190 

6°  20' 

2°     9'  58" 

6°20'  0" 

189.762 

7.178 

189.897 

1.930 

200 

6°  40' 

2°  23'  28" 

7°  0'  0" 

199.694 

8.339 

199.868 

2.234 

2K1- 

7°  00' 

2°  37    39" 

7°42'  0" 

209.612 

9.619 

209.832 

2.569 

220 

7°  20' 

2°  52'  28" 

8°26'  0" 

219.513 

11.022 

219.789 

2.934 

230 

7°  40' 

3°     1'  58" 

9<>i2/  0" 

229.395 

12.555 

229.738 

3.332 

240 

8°  00' 

3°  24'    6" 

10°  0'  0" 

239.255 

14.222 

239.677 

3.763 

25 
TABLE  III. 

°  *•• 

S 

d 
o 

Co-ordinates 

£«* 

tl 

£  > 

"o  T! 

n     £ 

Long 

Tft 

So 

O      ?-i 

§  j> 

"3    g> 

o     5 

Lati- 

Depar 

Chord. 

*f 

|Sr 

ft 

A 

Q      ^ 

tude. 

ture. 

as 

10 

0°  30 

0°     1'  30" 

0°     3'  0" 

10.000 

0.004 

10.000 

.003 

20 

1°  00' 

0°     3'  47" 

0°     9'  0" 

20.000 

0.022 

20.000 

.010 

4 

30 

1°  30' 

0°     6'  59" 

0°  18'  0" 

30.000 

0.063 

30.000 

.022 

40 

2°  00' 

0°  11'  15" 

0°  30'  0" 

40.000 

0.131 

40.000 

.044: 

50 

2°  30' 

0°  16'  30" 

0°  45'  0" 

49.999 

0.240 

49.999 

.080 

60 

3°  00' 

0°  22'  45" 

1°     3'0" 

59.998 

0.397 

59.999 

.126 

70 

3°  30' 

0°  30'     0" 

1°  24'  0" 

69.995 

0.611 

69.997 

.183 

80 

4°  00' 

0°  38'  15" 

1°  48'  0" 

79.991 

0.890 

79.996 

.262 

90 

4°  30' 

Oo  477  29" 

2°  15'  0" 

89.985 

1.243 

89.993 

.366 

100 

5°  00' 

0°  57'  44" 

2°  45'  0" 

99.975 

1.679 

99.989 

.487 

110 

5°  30' 

1°     8'  59" 

3°  18'  0" 

109.962 

2.207 

109.984 

.624 

120 

6°  00' 

1°  21'  14" 

3°  54'  0" 

119.942 

2.835 

119.975 

.793 

130 

6°  30' 

1°  34'  30" 

4°  33'  0" 

129.915 

3.572 

129.964 

1.001 

140 

7°  00' 

1°  48'  44" 

5°  15'  0" 

139.878 

4.426 

139.948 

1.231 

150 

7°  30' 

2°     3'  59" 

6°     0'  0" 

149.830 

5.406 

149.927 

1.480 

160 

8°  00' 

2°  20'  14" 

6°  48'  0" 

159.768 

6.521 

159.901 

1.776 

170 

8°  30' 

2°  37'  28" 

7°  39'  0" 

169.688 

7.778 

169.866 

2.130 

180 

9°  00' 

2°  55'  42" 

8°  33'  0" 

179.589 

9.187 

179.824 

2.491 

190 

9°  30' 

3°  14'  57" 

9°  30'  0" 

189.465 

10.756 

189.770 

2.889 

200 

10°  00' 

3°  35'  11" 

10°  30'  0" 

199.313 

12.492 

199.704 

3.342 

210 

10°  30' 

3°  56'  25" 

11°  33'  0" 

209.128 

14.405 

209.623 

3.855 

220 

11°  OO/ 

4°  18'  39'' 

12°  39'  0" 

218..  906 

16.501 

219.527 

4.399 

230 

11°  30' 

4°  41'  52" 

13°  48'  0" 

228.641 

18.V89 

229.411 

4.973 

240 

12°  OO' 

5°    6'    6" 

15°    O'O" 

238.327 

21.276 

239.275 

5.612 

26 
TABLE   IV. 

Length  of 
Curve. 

Degree  of 
Curve. 

Deflection 
Angle. 

Central 
Angle. 

Co-ordinates 

Long 
Chord. 

Corr'n  for 
Tang.Dist. 

Lati- 
tude. 

Depar 
ture. 

10 

0°  40' 

0°     2'    0" 

0°    4'   0" 

10.000 

0.006 

10.000 

.003 

20 

1°  20' 

0°     4'  59" 

0°  12'   0" 

20.000 

0.029 

20.000 

.012 

30 

2°  00' 

0°    9'  17" 

0°  24'   0" 

30.000 

0.081 

30.000 

.029 

40 

2°  40' 

0°  14'  57" 

0°  40'   0" 

39.999 

0.174 

39.999 

.058 

50 

3°  20' 

0°  22'     0" 

1°    0'   0" 

49.998 

0.320 

49.999 

.102 

60 

4°  00' 

0°  30'  19" 

1°  24'   0" 

59.996 

0.529 

59.998 

.162 

70 

4°  40' 

0°  39'  59" 

1°  52'   0" 

69.992 

0.814 

69.996 

.244 

FO 

5°  20' 

0°  50'  58" 

2°  24'   0" 

79.985 

1.186 

79.993 

.348 

90 

6°  00' 

1°     3'  18" 

3°    0'   0" 

89.974 

1.657 

89.989 

.479 

100 

6°  40' 

1°  16'  59" 

3°  40'   0" 

99.957 

2.239 

99.982 

.639 

110 

7°  20' 

1°  31'  58" 

4°  24'   0" 

109.932 

2.942 

109.971 

.831 

120 

8°  00' 

1°  48'  19" 

5°  12'   0" 

119.897 

3.779 

119.956 

1.057 

130 

8°  40' 

2°     5'  59" 

6°  04'  0" 

129.849 

4.761 

129.936 

1.321 

140 

9°  20' 

2°  24'  58" 

7°    O7  0" 

139.784 

5.898 

139.908 

1.625 

150 

10°  00' 

2°  45'  18" 

8°    0'  0" 

149.698 

7.204 

149.871 

1.972 

160 

10°  40' 

3°    6'  58" 

9°    4'  0" 

159.588 

8.688 

159.824 

2.365 

170 

11°  20' 

3°  29'  57" 

10°  12'   0" 

169.446 

10.361 

169.762 

2  805 

180 

12°  00' 

3°  54'  15" 

11°  24'  0" 

179.269 

12.235 

179.686 

3.296 

190 

12°  40' 

4°  19'  54" 

12°  40'   0" 

189.050 

14.320 

189.591 

3.841 

200 

13°  20' 

4°  46'  52" 

14°    0'  0" 

198.780 

16.626 

199.474 

4.439 

210 

14°  00' 

5°  15'    9" 

15°  24'  0" 

208.453 

19.163 

209.332 

5.094 

220 

14°  40' 

5°  44'  45" 

16°  52'  0" 

218.059 

21.942 

219.160 

5.809 

230 

15°  20' 

6°  15'  41" 

18°  24'   0" 

227.589 

24.971 

228.955 

6.585 

240  16°  00' 

6°  47'  56" 

20°    0'  0" 

237.033 

28.260 

238.711 

7.422 

Y     v*  Or    i  *a  r.  \ 

UNIVERSITY  ] 


27 
TABLE   V. 

I  Length  of 
1  Curve. 

Degree  of 
Curve. 

I    & 

§      §" 

<P          "^ 

14 

§  5 
0  ^ 

Co-ordinates 

Long 
Chord. 

!! 

Corr'n  for 
Tang.Dist,] 

Lati- 
tude. 

Depar 
ture. 

10 

0°  -60' 

0°     2'  30" 

0°    5'   0" 

10.000 

0.007 

10.000 

.004 

20 

1°  40' 

0°     6'  11" 

0°  15'   0" 

20.000 

0.036 

20.000 

.016 

30 

2°  30' 

0°  11'  41" 

0°  30'  0" 

29.999 

0.102 

29.999 

.037 

40 

3°  20' 

0°  18'  44" 

0°  50'  0" 

39.999 

0.218 

39.999 

.073 

50 

4°  10' 

0°  27'  30" 

1°  15'   0" 

49.997 

0.400 

49.998 

.131 

60 

5°  00' 

0°  37'  52" 

1°  45'  0" 

59.994 

0.661 

59.997 

.207 

70 

5°  60' 

0°  50'    0" 

2°  20'   0" 

69.987 

1.018 

69.994 

.305 

80 

6°  40' 

1°     3'  44" 

3°    0'   0" 

79.977 

1  .483 

79.990 

.436 

90 

7°  30' 

1°  19'  10" 

3°  45'   0" 

89.959 

2.072 

89.982 

.606 

100 

8°  20' 

1°  36'  14" 

4°  35'  0" 

99.933 

2.798 

99.972 

.806 

110 

9°  10' 

1°  54'  59" 

5°  30'  0" 

109.894 

3.677 

109.955 

1.038 

120 

10°  00' 

2°  15'  23" 

6°  30'  0" 

119.839 

4.722 

119.932 

1.321 

130 

10°  50' 

2°  37'  28" 

7°  35'  0" 

129.764 

5.948 

129.900 

1.659 

140 

11°  40' 

3°     V  13" 

8°  45'  0" 

139.662 

7.369 

139.856 

2.039 

150 

12°  30' 

3°  26'  37" 

10°    0'   0" 

149.529 

8.998 

149.799 

2.461 

160 

13°  20' 

3°  53'  41" 

11°  20'  0" 

159.356 

10.849 

159.725 

2.950 

170 

14°  10' 

4°  22'  24" 

12°  45'  0" 

169.136 

12.935 

169.630 

3.509 

180 

15°  00' 

4°  52'  46" 

14°  15'  0" 

178.860 

15.269 

179.510 

4.119 

190 

15°  50' 

5°  24'  48" 

15°  50'  0" 

188.517 

17.865 

189.361 

4.781 

200 

16°  40' 

5°  58'  29" 

17°  30'  0" 

198.097 

20.733 

199.179 

5.521 

210 

17°  30' 

6°  33'  49" 

19°  15'   0" 

207.587 

23.885 

208.95* 

6.344 

220 

18°  20' 

7°  10'  $1" 

21°    5'  0" 

216.974 

27.332 

218.689 

7.223 

230 

19°  10' 

7°  49'  23" 

23°    0'   0" 

226.243 

31.085 

228.368 

8.157 

240 

20°  00' 

8°  29'  39" 

25°    0'  0" 

235.379 

35.153 

237.989 

9.180 

28 
TABLE  VI. 


ii 

F 

10 

°  <£ 

«    > 

£  s 
Be 

p 

Deflection 
Angle. 

1     £ 
•£      M 
p      rt 

Co-ordinates 

Long 
Chord. 

Corr'n  for 
Tang.Dist. 

Lati- 
tude. 

Depar 
turs. 

1°  oo' 

0°    3'    0" 

0°     G'  0" 

10.000 

0.008 

10.000 

.004 

20 

2°  00' 

0°     7'  23" 

0°  18'  0" 

20.000 

0.043 

20.000 

.017 

30 

3°  00' 

0°  13'  58" 

0°  36'  0" 

29.999 

0.122 

29.999 

.043 

40 

4°  00' 

0°  22'  26" 

1°     0'  0" 

39.998 

0.261 

39.999 

.086 

50 

5°  00' 

0°  33'    0" 

1°  30'  0" 

49.996 

0.480 

49.998 

.153 

60 

6°  00' 

0°  43'  30" 

2°     6'  0" 

59.991 

0.794 

59.996 

.244 

70 

7°  00' 

0°  59'  58" 

2°  48'  0" 

69.982 

1.221 

69.992 

.366 

80 

8°  00' 

1°  16'  28" 

3°  36'  0" 

79.966 

1.779 

79.985 

.523 

90 

9°  00' 

1°  35'     0" 

4°  30'  0" 

89.941 

2.486 

89.975 

.719 

100 

10°  00' 

1°  55'  28" 

5°  30'  0" 

99.903 

3.357 

99.959 

.958 

110 

11°  00' 

2°  17'  58" 

6°  36'  0" 

109.848 

4.411 

109.936 

1.245 

120 

12°  00' 

2°  42'  29" 

7°  48'  0" 

119.769 

5.665 

119.903 

1  584 

130 

13°  00' 

3°    8'  57" 

9°  06'  0" 

129.660 

7.134 

129.856 

1.978 

140 

14°  00' 

3°  37'  26" 

10°  30'  0" 

139.514 

8.836 

139.793 

2.431 

150 

15°  00' 

4°     V  55" 

12°  00'  0" 

149.322 

10.787 

149.711 

2.947 

160 

16°  00' 

4°  40'  23" 

13°  36'  0" 

159.073 

13.003 

159.603 

3.531 

170 

17°  00' 

5°  14'  49" 

15°  18'  0" 

168.757 

15.498 

169.467 

4.183 

180 

18°  00' 

5°  51'  15" 

17°  06'  0" 

178.360 

18.288 

179.295 

4.908 

190 

19°  00' 

6°  29'  39" 

19°     0'  0" 

187.868 

21.386 

189.081 

5.706 

200 

20°  00' 

7°  10'     2" 

21°     0'  0" 

197.265 

24.806 

198.818 

6.582 

210 

21°  00' 

7°  52'  24" 

23°     6'0" 

206.534 

28.561 

208.499 

7.538 

220 

22°  00' 

8°  36'  42" 

25°  18'  0" 

215.655 

32.660 

218.114 

8.571 

230 

23°  00' 

9°  22'  58" 

27°  36'  0" 

224.608 

37.114 

227.654 

9.683 

240/24°  00' 

10°  11'  10" 

30°     0'  0" 

233.371 

41.932 

237.108 

10.874 

TABLE  OF  TANGENT  DISTANCES 

FOB 

CIRCULAR  RAILROAD  CURVES. 


This  table  contains  the  tangent  distances 
to  a  1  degree  curve  for  every  two  minutes 
of  intersection  angle  to  90  degrees. 

The  tangent  distance  to  any  other  de- 
gree of  curve  may  be  determined  by  divid- 
ing the  number  corresponding  to  the  inter- 
section angle  by  the  degree  of  curve. 


.     OF  THF 

UNIVERSITY  ) 


31 


t 

0° 

1° 

2° 

3° 

/ 

0 

0.00 

50.00 

100.00 

150.07 

0 

2 

1.67 

51.67 

101.67 

151.74 

2 

4 

3.33 

53.33 

103  34 

153.41 

4 

6 

5.00 

55.00 

105.01 

155.08 

6 

8 

6.67 

56.67 

106.68 

156  .  75 

8 

10 

8.33 

58.33 

108.35 

158.42 

10 

12 

10.00 

60.00 

110.02 

160.09 

12 

14 

11.67 

61.67 

111.69 

161.76 

14 

16 

13.33 

63.33 

113.36 

163.43 

16 

18 

15.00 

65.00 

115.02 

165.09 

18 

20 

16.67 

66.67 

116.69 

166.76 

20 

22 

18.33 

68.33 

118.36 

168.43 

22 

24 

20.00 

70.00 

120.03 

170.10 

24 

26 

21.67 

71.67 

121.70 

171.77 

26 

28 

23.33 

73.33 

123.37 

173.44 

28 

30 

25.00 

75.00 

125.03 

175.10 

30 

32 

26.67 

76.67 

126.70 

176.72 

32 

34 

28.33 

78.33 

128.37 

178.39 

34 

36 

30.00 

80.00 

130.04 

180.06 

36 

38 

31.67 

81.67 

131.71 

181.73 

38 

40 

33.33 

83.33 

133.38 

183.40 

40 

42 

35.00 

85.00 

135.05 

185.07 

42 

44 

36.67 

86.6^ 

136.72 

186.74 

44 

46 

38.33 

88.33 

138.38 

188.40 

46 

48 

40.00 

90.00 

140.05 

190.07 

48 

50 

41.67 

91.67 

141.72 

191.74 

50 

52 

43.33 

93.33 

143.39 

193.41 

52 

54 

45.00 

95.00 

145.06 

195.08 

54 

56 

46.67 

96.67 

146  .  73 

196.75 

56 

58 

48.33 

98.33 

148.40 

198.42 

58 

60 

50.00 

100.00 

150.07 

200.09 

60 

32 


/ 

4° 

5° 

6°    7° 

/ 

0 

200.09 

250.17 

300.30 

350.44 

0 

2 

201  .  76 

251.84 

301.97 

352.11 

2 

4 

203.43 

253.51 

303  .  64 

353.79 

4 

6 

205.10 

255.18 

305.31 

355.46 

6 

8 

206.77 

256  .  85 

306.98 

357.13 

8 

10 

208.44 

258.52 

308.65 

358.81 

10 

12 

210.11 

260.20 

310.32 

360.48 

12 

14 

211.77 

261.86 

311.99 

362.15 

14 

16 

213.45 

263.54 

313.66 

363.83 

16 

18 

215.11 

265.20 

315.33 

365.50 

18 

20 

216.78 

266.87 

317.00 

367.17 

20 

22 

218.45 

268.54 

318.67 

368.85 

22 

24 

220.12 

270.21 

320.34 

370.52 

24 

26 

221.79 

271.88 

322.01 

372.19 

26 

28 

223.46 

273.54 

323  .  68 

373.86 

28 

30 

225.13 

275.21 

325.35 

375.54 

30 

32 

226.80 

276.88 

327.02 

377.22 

32 

34 

228.47 

278.55 

328.69 

378.89 

34 

36 

230.14 

280.23 

330.37 

380.57 

36 

38 

231.81 

281.90 

332.04 

382.24 

38 

40 

233.48 

283.57 

333.71 

383.92 

40 

42 

235.15 

285.24 

335.38 

385.60 

42 

44 

236.82 

286.91 

337.05 

387.27 

44 

46 

238.48 

288.59 

338.73 

388.95 

46 

48 

240.15 

290.26 

340.40 

390.62 

48. 

50 

241.82 

291.93 

342.07 

392.30 

50 

52 

243.49 

293.60 

343.74 

393.98 

52 

54 

245.16 

295.27 

345.41 

395.65 

54 

56 

246.83 

296.95 

347.08 

397.33 

56 

58 

248.50 

298.62 

348.76 

399.01 

58 

60 

250.17 

300.30 

350.44 

400.70 

60 

33 


t 

8° 

9° 

10° 

11° 

t 

~0 

400.70 

450.95 

501.32 

551.74 

0 

2 

402.37 

452.63 

503.00 

553.42 

2 

4 

404.05  454.31 

504.68 

555.10 

4 

6 

405  .  72 

455.98 

506.36 

556.78 

6 

8 

407.39  457.66 

508.04 

558.46 

8 

10 

409.06  459.34 

509.72 

560.14 

10 

12 

410.74  461.02 

511.40 

561.82 

12 

14 

412.  41  j  462.  70 

513.08 

563.50 

14 

16 

414.08  464.37 

514.76 

565.18 

16 

18 

415.75 

466.05 

516.44 

566.86 

18 

20 

417.43 

467.73 

518.12 

568.54 

20 

22 

419.10 

469.41 

519.80 

570.22 

22 

24 

420.77 

471.08 

521.48 

571.90 

24 

26 

422.45 

472.76 

523.16 

573.58 

26 

28 

424.12 

474.43 

524.85 

575.27 

28 

30 

425.79 

476.10 

526.53 

576.95 

30 

32 

427.47 

477.78 

528.21 

578.63 

32 

34 

429.15 

479.46 

529.89 

580.32 

34 

36 

430.82 

481.14 

531.57 

582.00 

36 

38 

432.50 

482.83 

533.25 

583.69 

38 

40 

434.18 

484.51 

534.93 

585.37 

40 

42 

435.86 

486.19 

536.61 

587.05 

42 

44 

437.54 

487.87 

538.29 

588.74 

44 

46 

439.21 

489.56 

539.97 

590.42 

46 

48 

440.89 

491.24 

541.65 

592.11 

48 

50 

442.57 

492.92 

543.33 

593.79 

50 

52 

444.25 

494.60 

545.01 

595.47 

52 

54 

445.93 

496.28 

546.69 

597.16 

54 

56 

447.60 

497.96 

548.37 

598.84 

56 

58 

449.28 

499.65 

550.06 

600.53 

58 

60 

450.95 

501.32 

551.74 

602.22 

60 

34 


/ 

12° 

13° 

14° 

15° 

/ 

0 

602.22 

652.87 

703.53 

754.35 

0 

2 

603.91 

654.56 

705.23 

756.05 

2 

4 

605.60 

656.25 

706.92 

757.74 

4 

6 

607.28 

657.93 

708.62 

759.44 

6 

8 

608.97 

659.62 

710.31 

761.13 

8 

10 

610.66 

661.31 

712.01 

762.83 

10 

12 

612.35 

663.00 

713.71 

764.53 

12 

14 

614.04 

664.69 

715.40 

766.22 

14 

16 

615.72 

666.37 

717.10 

767.92 

16 

18 

617.41 

668.06 

718.79 

769.61 

18 

20 

619.10 

669.75 

720.49 

771.31 

20 

22 

620.79 

671.44 

722.20 

773.01 

22 

24 

622.48 

673.13 

723.89 

774.70 

24 

26 

624.16 

674.81 

725.59 

776.40 

26 

28 

625.85 

676.51 

727.28 

778.09 

28 

30 

627.55 

678.20 

728.97 

779.79 

30 

32 

629.24 

679.89 

730.66 

781.49 

32 

34 

630-93 

681.58 

732.35 

783.19 

34 

36 

632.61 

683.26 

734.05 

784.89 

36 

38 

634.30 

684.95 

735.74 

786.59 

38 

40 

635.99 

686.64 

737.43 

788.29 

40 

42 

637.68 

688.33 

739.12 

789.99 

42 

44 

639.37 

690.02 

740.81 

791.69 

44 

46 

641.  C5 

691.70 

742.51 

793.39 

46 

48 

642.74 

693.39 

744.20 

795.09 

48 

50 

644.43 

695.08 

745.89 

796.79 

50 

52 

646.12 

696.77 

747.58 

798.49 

52 

54 

647.81 

698.46 

749.27 

800.19 

54 

56 

649.49 

700.14 

750.97 

801.89 

56 

58 

651.18 

701.83 

752.66 

803.59 

58 

60 

652.87 

703.53 

754.35 

805.29 

60 

35 


/ 

16° 

17° 

18° 

19° 

/ 

9 

805.29 

856.35 

907.52 

958.86 

0 

2 

806.99 

858.05 

909.23 

960.57 

2 

4 

808.64 

859.76 

910.94 

962.30 

4 

6 

810.39 

861.46 

912.65 

964.00 

6 

8 

812.09 

863  .  16 

914.36 

965.72 

8 

10 

813.79 

864.87 

916.07 

967.43 

10 

12 

815.49 

866.57 

917.78 

969.15 

12 

14 

817.19 

868.27 

919.49 

970.86 

14 

16 

818.89 

869.98 

921.20 

972.58 

16 

18 

820.59 

871.68 

922.91 

974.29 

18 

20 

822.29 

873.38 

924.63 

976.01 

20 

22 

823  .  99 

875.09 

926.34 

977.72 

22 

24 

825.69 

876.79 

928.05 

979.44 

24 

26 

827.39 

878.49 

929.76 

981.15 

26 

28 

829.09 

880.20 

931.47 

982.86 

28 

30 

830.79 

881.90 

933.18 

984.58 

30 

32 

832.49 

883.61 

934.89 

986.30 

32 

34 

834.20 

885.32 

936.60 

988.02 

34 

36 

835.90 

887.02 

938.32 

989.74 

36 

38 

837.61 

888.73 

940.03 

991.46 

38 

40 

839.31 

890.44 

941  .  74 

993.18 

40 

42 

841.01 

892.15 

943.45 

994.90 

42 

44 

842.72 

893.86 

945.16 

996.62 

44 

46 

844.42 

895.56 

946.88 

998.34 

46 

48 

846.13 

897.27 

948.59 

1000.0 

48 

50 

847.83 

898.98 

950.30 

1001.8 

50 

52 

849.53 

900.69 

952.01 

1003.5 

52 

54 

851.24 

902.40 

953.72 

1005.2 

54 

56 

852.94 

904.10 

955.44 

1006.9 

56 

58 

854.65 

905.81 

957.15 

1008.6 

58 

60 

856.35 

907.52 

958.86 

1010.4 

60 

36 


/ 

20° 

21° 

22° 

23° 

/ 

0 

1010.4 

1062.0 

1113.8 

1165.8 

~cT 

2 

1012.1 

1063.7 

1115.5 

1167.5 

2 

4 

1013.8 

1065.4 

1117.3 

1169.2 

4 

6 

1015.5 

1067.2 

1119.0 

1171.0 

6 

8 

1017.2 

1068.9 

1120.7 

1172.7 

8 

10 

1019.0 

1070.6 

1122.4 

1174.4 

10 

12 

1020.7 

1072.4 

1124.2 

1176.2 

12 

14 

1022.4 

1074.1 

1125.9 

1177.9 

14 

16 

1024.1 

1075.8 

1127.6 

1179.7 

16 

18 

1025.8 

1077.5 

1129.4 

1181.4 

18 

20 

1027.6 

1079.3 

1131.1 

1183.1 

20 

22 

1029.3 

1081.0 

1132.8 

1184.9 

22 

24 

1031.0 

1082.7 

1134.6 

1186.6 

24 

26 

1032.7 

1084.4 

1136.3 

1188.4 

26 

28 

1034.4 

1086.2 

1138.0 

1190.1 

28 

30 

1036.1 

1087.9 

1139.7 

1191.8 

30 

32 

1037.9 

1089.6 

1141.5 

1193.6 

32 

34 

1039.6 

1091.3 

1143.2 

1195.3 

34 

36 

1041.3 

1093.1 

1144.9 

1197.1 

36 

38 

1043.0 

1094.8 

1146.7 

1198.8 

38 

40 

1044.8 

1096.5 

1148.4 

1200.5 

40 

42 

1046.5 

1098.3 

1150.1 

1202.3 

42 

44 

1048.2 

1100.0 

1151.9 

1204.0 

44 

46 

1049.9 

1101.7 

1153.6 

1205.8 

46 

48 

1051.7 

1103.4 

1155.4 

1207.5 

48 

50 

1053.4 

1105.2 

1157.1 

1209.2 

50 

52 

1055.1 

1106.9 

1158.8 

1211.0 

52 

54 

1056.8 

1108.6 

1160.6 

1212.7 

54 

56 

1058.6 

1110.3 

1162.3 

1214.5 

56 

58 

1060.3 

1112.1 

1164.0 

1216.2 

58 

60 

1062,OJ1113.8 

1165,8 

1218.0 

60 

37 


' 

24° 

25° 

26° 

27° 

' 

0 

1218.0 

1270.3 

1322.9 

1375.6 

0 

2 

1219.7 

1272.0 

1324.6 

1377.4 

2 

4 

1221.4 

1273.8 

1326.4 

1379.2 

4 

6 

1223.2 

1275.5 

1328.1 

1380.9 

6 

8 

1224.9 

1277.3 

1329.9 

1382.7 

8 

10 

1226.7 

1279.0 

1331.6 

1384.5 

10 

12 

1228.4 

1280.8 

1333.4 

1386.2 

12 

14 

1230.2 

1282.5 

1335.2 

1388.0 

14 

16 

1231.9 

1284.3 

1336.9 

1389.8 

16 

18 

1233.6 

1286.1 

1338.7 

1391.5 

18 

20 

1235.4 

1287.8 

1340.4 

1393.3 

20 

22 

1237.1 

1289.6 

1342.2 

1395.0 

22 

24 

1238.9 

1291.3 

1343.9 

1396.8 

24 

26 

1240.6 

1293.1 

1345  .  7 

1398.6 

26 

28 

1242.4 

1294.8 

1347.4 

1400.3 

28 

30 

1244.1 

1296.6 

1349.2 

1402.1 

30 

32 

1245.8 

1298.3 

1351.0 

1403.9 

32 

34 

1247.6 

1300.1 

1352.7 

1405.6 

34 

36 

1249.3 

1301.8 

1354.5 

1407.4 

36 

38 

1251.1 

1303.6 

1356.2 

1409.2 

38 

40 

1252.8 

1305.3 

1358.0 

1410.9 

40 

42 

1254.6 

1307.1 

1359.8 

1412.7 

42 

44 

1256.3 

1308.8 

1361.5 

1414.5 

44 

46 

1258.1 

1310.6 

1363.3 

1416.3 

46 

48 

1259.8 

1312.4 

1365.1 

1418.0 

48 

50 

1261.5 

1314.1 

1366.8 

1419.8 

50 

52 

1263.3 

1315.9 

1368.6 

1421.6 

52 

54 

1265.0 

1317.6 

1370.4 

1423.3 

54 

56 

1266.8 

1319.4 

1372.1 

1425.1 

56 

58 

1268.5 

1321.1 

1373.9 

1426.9 

58 

60 

1270.3 

1322.9 

1375.6 

1428.6 

60 

38 


/ 

28° 

29° 

30° 

31° 

/ 

0 

1428.6 

1481  .  9 

1535.3 

1589.0 

0 

2 

1430.4 

1483  .  7 

1537.1 

1590.8 

2 

4 

1432.2 

1485.4 

1538.9 

1592.6 

4 

6 

1434.0 

1487.2 

1540.7 

1594.4 

6 

8 

1435  .  7 

1489.0 

1542.5 

1596.2 

8 

10 

1437.5 

1490.8 

1544.3 

1598.0 

10 

12 

1439.3 

1492.6    1546.0 

1599.8 

12 

14 

1441.1 

1494.3 

1547.8 

1601.6 

14 

16 

1442.8 

1496.1 

1549.6 

1603.4 

16 

18 

1444.6 

1497.9 

1551.4 

1605.2 

18 

20 

1446.4 

1499.7 

1553.2 

1607.1 

20 

22 

1448.2 

1501.5 

1555.0 

1608.8122 

24 

1449.9 

1503.2 

1556.8 

1610.1   24 

26 

1451.7 

1505.0 

1558.6 

1612.4  ,26 

28 

1453.5 

1506.8 

1560.4 

i614.fc   28 

30 

1455.2 

1508.6 

1562.2 

161  6.  (    30 

32 

1457.0 

1510.4 

1564.0 

1617.*    32 

34 

1458.8 

1512.1 

1565.7 

1619.  (    34 

36 

1460.6 

1513.9 

1567.5 

1621.4 

36 

38 

1462.3 

1515.7 

1569.3 

1623.2 

38 

40 

1464.1 

1517.5 

1571.1 

1625.  I 

40 

42 

1465.9 

1519.3 

1572.9 

1626.8 

42 

44 

1467.7 

1521.0 

1574.7 

1628.6 

44 

46 

1469.5 

1522.8 

1576.5 

1630.5 

46 

48 

1471.2 

1524.6 

1578.3 

1632.3 

48 

50 

1473.0 

1526.4 

1580.1 

1634.1   50 

52 

1474.8 

1528.2 

1581.9 

1635.9   52 

54 

1476.6 

1530.0 

1583.7 

1637.7 

54 

56 

1478.3 

1531.^ 

1585.5 

1639.5 

56 

58 

1480.1 

1533.5 

1587.2 

1641.3 

58 

60 

1481.9 

1535.3 

1589.0 

1643.1 

60 

39 


/ 

32° 

35° 

34° 

35° 

/ 

0 

1643.1 

1697.3 

1751.8 

1806.7 

0 

2 

1644.9 

1699.1 

1753.7 

1808.5 

2 

4 

1646.7 

1700.9 

1755.5 

1810.3 

4 

6 

1648.5 

1702.7 

1757.3 

1812.2 

6 

8 

1650.3 

1704.5 

1759.1 

1814.0 

8 

10 

1652.1 

1706.4 

1761.0 

1815.8 

10 

12 

1653.9 

1708.2 

1762.8 

1817.7 

12 

14 

1655.7 

1710.0 

1764.6 

1819.5 

14 

16 

1657.5 

1711.8 

1766.4 

1821.3 

16 

18 

1659.3 

1713.6 

1768.3 

1823.2 

18 

20 

1661.1 

1715.5 

1770.1 

1825.0 

20 

22 

1662.9 

1717.3 

1771.9 

1826.8 

22 

24 

1664.7 

1719.1 

1773.7 

1828.7 

24 

26 

1666.5 

1720.9 

1775.6 

1830.5 

26 

28 

1668.3 

1722.7 

1777.4 

1832.3 

28 

30 

1670.1 

1724.6 

1779.2 

1834.2 

30 

32 

1671.9 

1726.4 

1781.0 

1836.0 

32 

34 

1673.7 

1728.2 

1782.9 

1837.8 

34 

36 

1675.5 

1730.0 

1784.7 

1839.7 

36 

38 

1677.4 

1731.8 

1786.5 

1841.5 

38 

40 

1679.2 

1733.6 

1788.4 

1843.4 

40 

42 

1681.0 

1735.5 

1790.2 

1845.2 

42 

44 

1682.8 

1737.3 

1792.0 

1847.1 

44 

46 

1684.6 

1739.1 

1793.9 

1848.9 

46 

48 

1686.4 

1740.9 

1795.7 

1850.7 

48 

50 

1688.2 

1742.7 

1797.5 

1852.6 

50 

52 

1690.0 

1744.6 

1799.3 

1854.4 

52 

54 

1691.8 

1746.4 

1801.2 

1856.3 

54 

56 

1693.7 

1748.2 

1803.0 

1858.1 

56 

58 

1695.5 

1750.0 

1804.8 

1859.9 

58 

60 

1697.3 

1751.8 

1806.7 

1861.8 

60 

40 


/ 

36° 

37° 

38* 

39° 

/ 

0 

1861.8 

1917.3 

1973.0 

2029.1 

0 

2 

1863.6 

1919.1 

1974.9 

2031.0 

2 

4 

1865.5 

1921.0 

1976.7 

2032.9 

4 

6 

1867.3 

1922.8 

1978.6 

2034.7 

6 

8 

1869.2 

1924.7 

1980.5 

2036.6 

8 

10 

1871.0 

1926.5 

1982.3 

2038.5 

10 

12 

1872.9 

1928.4 

1984.2 

2040.4 

12 

14 

1874.7 

1930.2 

1986.1 

2042.3 

14 

16 

1876.5 

1932.1 

1987.9 

2044.1 

16 

18 

1878.4 

1933.9 

1989.8 

2046.0 

18 

20 

1880.2 

1935.8 

1991.7 

2047.9 

20 

22 

1882.1 

1937.6 

1993.6 

2049.8 

22 

24 

1883.9 

1939.5 

1995.4 

2051.7 

24 

26 

1885.8 

1941.3 

1997.3 

2053.5 

26 

28 

1887.6 

1943.2 

1999.2 

2055.4 

28 

30 

1889.5 

1945.0 

2001.0 

2057.3 

30 

32 

1891.3 

1946.9 

2002.9 

2059.2 

32 

34 

1893-2 

1948.8 

2004.8 

2061.1 

34 

36 

1895.0 

1950.6 

2006.6 

2063.0 

36 

38 

1896.9 

1952.5 

2008.5 

2064.8 

38 

40 

1898.7 

1954.4 

2010.4 

2066.7 

40 

42 

1900.6 

1956.2 

2012.3 

2068.6 

42 

44 

1902.4 

1958.1 

2014.1 

2070.5 

44 

46 

1904.3 

1960.0 

2016.0. 

2072.4 

46 

48 

1906.1 

1961.8 

2017.9 

2074.2 

48 

50 

1908.0 

1963.7 

2019.7 

2076.1 

50 

52 

1909.8 

1965.5 

2021.6 

2078.0 

52 

54 

1911.7 

1967.4 

2023.5 

2079.9 

54 

56 

1913.5 

1969.3 

2025.4 

2081.8 

56 

58 

1915.4 

1971.1 

2027.2 

2083  .  7 

58 

60 

1917.3 

1973.0 

2029.1 

2085.5 

60 

4:1 


'  |  40° 

41° 

42° 

43° 

/ 

0  j  2085!  5 

2142.3 

2199.5 

2257.1 

0 

2 

2087.4 

2144.2 

2201.4 

2259.0 

2 

4 

2089.3 

2146.1 

2203.3 

2261.0 

4 

6 

2091.2 

2148.0 

2205.3 

2262.9 

6 

8 

2093.1 

2149.9 

2207.2 

2264.8 

8 

10 

2095.0 

2151.9 

2209.1 

2266.7 

10 

12 

2096.9 

2153.8 

2211.0 

2268.7 

12 

14 

2098.8 

2155.7 

2212.9 

2270.0 

14 

16 

2100.7 

2157.6 

2214.9 

2272.5 

16 

18 

2102.6 

2159.5 

2216.8 

2274.5 

18 

20 

2104.5 

2161.4 

2218.7 

2276.4 

20 

22 

2106.3 

2163.3 

2220.6 

2278.3 

22 

24 

2108.2 

2165.2 

2222.5 

2280.2 

24 

26 

2110.1 

2167.1 

2224.4 

2282.2 

26 

28 

2112.0 

2169.0 

2226.4 

2284.1 

28 

30 

2113.9 

2170.9 

2228.3 

2286.0 

30 

32 

2115.8 

2172.8 

2230.2 

2288.0 

32 

34 

2117.7 

2174.7 

2232.1 

2289.9 

34 

36 

2119.6 

2176.6 

2234.0 

2291.8 

36 

38 

2121.5 

2178.5 

2236.0 

2293.8 

38 

40 

2123.4 

2180.4 

2237.9 

2295.7 

40 

42 

2125.3 

2182.4 

2239.8 

2297.7 

42 

44 

2127.2 

2184.3 

2241.7 

2299.6 

44 

46 

2129.1 

2186.2 

2243  .  6 

2301.5 

46 

48 

2131.0 

2188.1 

2245  .  6 

2303.5 

48 

50 

2132.9 

2190.0 

2247.5 

2305.4 

50 

52 

2134.7 

2191.9 

2249.4 

2307.3 

52 

54 

2136.6 

2193.8 

2251.3 

2309.3 

54 

56 

2138.5 

2195.7 

2253.3 

2311.2 

56 

58 

2140.4 

2197.6 

2255.2 

2313.1 

58 

60 

2142.3 

2199.5 

2257.1 

2315.1 

60 

42 


' 

44° 

45° 

46° 

47° 

' 

0 

2315.1 

2373.4 

2432.2 

2491.5 

0 

2 

2317.0 

2375.4 

2434.2 

2493.4 

2 

4 

2319.0 

2377.3 

2436.1 

2495.4 

4 

6 

2320.9 

2379.3 

2438.1 

2497.4 

0 

8 

2322.8 

2381.2 

2440.1 

2499.4 

8 

10 

2324.8 

2383.2 

2442.1 

2501.4 

10 

12 

2326.7 

2385.2 

2444.0 

2503.4 

12 

14 

2328.7 

2387.1 

2446.0 

2505.4 

14 

16 

2330.6 

2389.1 

2448.0 

2507.3 

16 

18 

2332  .  6 

2391.0 

2449  .  9 

2509.3 

18 

20 

2334.5 

2393.0 

2451.9 

2511.3 

20 

22 

2336.4 

2394.9 

2453.9 

2513.3 

22 

24 

2338.4 

2396.9 

2455.9 

2515.3 

24 

26 

2340.3 

2398.8 

2457.8 

2517.3 

26 

28 

2342.3 

2400.8 

2459.8 

2519.3 

28 

30 

2344.2 

2402.8 

2461.8 

2521.2 

30 

32 

2346.1 

2404.7 

2463.8 

2523.2 

32 

34 

2348.1 

2406.7 

2465.7 

2525.2 

34 

36 

2350.0 

2408.6 

2467.7 

2527.2 

36 

38 

2352.0 

2410.6 

2469.7 

2529.2 

38 

40 

2353.9 

2412.6 

2471  .  7 

2531.2 

40 

42 

2355.9 

2414.5 

2473.6 

2533.2 

42 

44 

2357.8 

2416.5 

2475.6 

2535.2 

44 

46 

2359.8 

2418.5 

2477.6 

2537.2 

46 

48 

2361.7 

2420.4 

2479.6 

2539.2 

48 

50 

2363.7 

2422  .4 

2481.6 

2541  .'2 

50 

52 

2365.6 

2424.4 

2483.5 

2543.1 

52 

54 

2367.6 

2426.3 

2485.5 

2545.1 

54 

56 

2369.5 

2428.3 

2487.5 

2547.1 

56 

58 

2371.5 

2430.2 

2489.5 

2549.1 

58 

60 

2373.4 

2432.2 

2491.5 

2551.1 

60 

43 


r 

48° 

49° 

50° 

51° 

/ 

0 

2551.1 

2611.3 

2671.9 

2733.0 

0 

2 

2553.1 

2613.3 

2673.9 

2735.1 

2 

4 

2555,1 

2615.3 

2676.0 

2737.1 

4 

6 

2557.1 

2617.3 

2678.0 

2739.2 

6 

8 

2559.1 

2619.3 

2680.0 

2741.2 

8 

10 

2561.1 

2621.4 

2682.1 

2743.3 

10 

12 

2563.1 

2623.4 

2684.1 

2745.3 

12 

14 

2565.1 

2625.4 

2686.1 

2747.4 

14 

16 

2567.1 

2627.4 

2688.2 

2749.4 

16 

18 

2569.1 

2629.4 

2690.2 

2751.5 

18 

20 

2571.1 

2631.4 

2692.3 

2753.5 

20 

22 

2573.1 

2633.5 

2694.3 

2755.6 

22 

24 

2575.1 

2635.5 

2696.3 

2757.7 

24 

2(j 

2577.1 

2637.5 

2698.4 

2759.7 

26 

28 

2579.1 

2639.5 

2700.4 

2761.8 

28 

30 

2581.1 

2641.5 

2702.4 

2763.8 

30 

32 

2583.1 

2643.5 

2704.5 

2765.9 

32 

34 

2585.1 

2645.6 

2706.5 

2767.9 

34 

36 

2587.2 

2647.6 

2708.6 

2770.0 

36 

38 

2589.2 

2649.6 

2710.6 

2772.0 

38 

40 

2591.2 

2651.6 

2712.6 

2774.1 

40 

4:2 

2593.2 

2653.7 

2714.7 

2776.2 

42 

44 

2595.2 

2655.7 

2716.7 

2778.2 

44 

46 

2597.2 

2657.7 

2718.8 

2780.3 

46 

48 

2599.2 

2659.7 

2720.8 

2782.3 

48 

50 

2601.2 

2661.8 

2722.8 

2784.4 

50 

52 

2603.2 

2663.8 

2724.9 

2786.4 

52 

54 

2605.2 

2665.8 

2726.9 

2788.5 

54 

56 

2607.2 

2667.8 

2729.0 

2790.6 

56 

58 

2609.3 

2669.9 

2731.0 

2792.6 

58 

60 

2611.3 

2671.9 

2733.0 

2794.7 

60 

44 


/ 

52° 

53° 

54° 

55° 

/ 

0 

2794.7 

2856.9 

2919.5 

2982.8 

0 

2 

2796.8 

2858.9 

2921.6 

2984.9 

2 

4 

2798.8 

2861.0 

2923.8 

2987.1 

4 

6 

2800.9 

2863.1 

2925.9 

2989.2 

6 

8 

2803.0 

2865.2 

2928.0 

2991.3 

8 

10 

2805.0 

2867.3 

2930.1 

2993.4 

10 

12 

2807.1 

2869.4 

2932.2 

2995.5 

12 

14 

2809.2 

2871.5 

2934.3 

2997.7 

14 

16 

2811.2 

2873.5 

2936.4 

2999.8 

16 

18 

2813.3 

2875.6 

2938.5 

3001.9 

18 

20 

2815.4 

2877.7 

2940.6 

3004.0 

20 

22 

2817.4 

2879.8 

2942.7 

3006.2 

22 

24, 

2819.5 

2881.9 

2944.8 

3008.3 

24 

26 

2821.6 

2884.0 

2946.9 

3010.4 

26 

23 

2823.6 

2886.1 

2949.0 

3012.5 

28 

30 

2825.7 

2888.1 

2951.1 

3014.7 

30 

32 

2827.8 

2890.2 

2953.2 

3016.8 

32 

34 

2829.8 

2892.3 

2955.3 

3018.9 

34 

36 

2831.9 

2894.4 

2957.5 

3021.1 

36 

38 

2834.0 

2896.5 

2959.6 

3023.2 

38 

40 

2836.1 

2898.6 

2961.7 

3025.3 

40 

42 

2838.2 

2900.7 

2963.8 

3027.5 

42 

44 

2840.2 

2902.8 

2965.9 

3029.6 

44 

46 

2842.3 

2904.9 

2968.0 

3031.7 

46 

48 

2844.4 

2907.0 

2970.1 

3033.8 

48 

50 

2846.5 

2909.1 

2972.2 

3036.0 

50 

52 

2848.5 

2911.2 

2974.4 

3038.1 

52 

54 

2850.6 

2913.3 

2976.5 

3040.2 

54 

56 

2852.7 

2915.4 

2978.6 

3042.4 

56 

58 

2854.8 

2917.5 

2980.7 

3044.5 

58 

60 

2856.9 

2919.5 

2982.8 

3046.6 

60 

45 


/ 

56° 

57°    58° 

59° 

/ 

0 

3046.6 

3111.1 

3176.1 

3241.9 

0 

2 

3048.8 

3113.3 

3178.3 

3244.1 

2 

4 

3050.9 

3115.4 

3180.5 

3246.3 

4 

6 

3053.1 

3117.6 

3182.7 

3248.5 

6 

8 

3055.2 

3119.7 

3184.9 

3250.7 

8 

10 

3057.4 

3121.9 

3187.1 

3252.9 

10 

12 

3059.5 

3124.1 

3189.2 

3255.1 

12 

14 

3061.6 

3126.2 

3191.4 

3257.3 

14 

16 

3063.8 

3128.4 

3193.6 

3259.5 

16 

18 

3065.9 

3130.6 

3195.8 

3261.7 

18 

20 

3068.1 

3132.7 

3198.0 

3263.9 

20 

22 

3070.2 

3134.9 

3200.2 

3266.1 

22 

24 

3072.4 

3137.0 

3202.4 

3268.3 

24 

26 

3074.5 

3139.2 

3204.5 

3270.5 

26 

28 

3076.6 

3141.4 

3206.7 

3272.7 

28 

30 

3078.8 

3143.5 

3208.9 

3274.9 

30 

32 

3080.9 

3145.7 

3211.1 

3277.1 

32 

34 

3083.1 

3147-9 

3213.3 

3279.4 

34 

36 

3085.2 

3150.0 

3215.5 

3281.6 

36 

38 

3087.4 

3152.2 

3217.7 

3283.8 

38 

40 

3089.6 

3154.4 

3219.9 

3286.0 

40 

42 

3091.7 

3156.6 

3222.1 

3288.2 

42 

44 

3093.9 

3158.7 

3224.3 

3290.5 

44 

46 

3096.0 

3160.9 

3226.5 

3292.7 

46 

48 

3098.2 

3163.1 

3228.7 

3294.9 

48 

50 

3100.3 

3165.3 

3230.9 

3297.1 

50 

52 

3102.5 

3167.4 

3233.1 

3299.3 

52 

54 

3104.6 

3169.6 

3235.3 

3301.5 

54 

56 

3106.8 

3171.8 

3237.5 

3303.8 

56 

58 

3108.9 

3174.0 

3239.7 

3306.0 

58 

60 

3111.1  3176.1 

3241  .  9 

3308.2 

60 

/ 

60°   61° 

62° 

63° 

f 

0 

3308.2 

3375.2 

3442.9 

3511.3 

0 

2 

3310.4 

3377.4 

3445.2 

3513.6 

2 

4 

3312.7 

3379.7 

3447.5 

3515.9 

4 

6 

3314.9 

3381.9 

3449  .  7 

3518.2 

6 

8 

3317.1 

3384.2 

3452.0 

3520.5 

8 

10 

3319.3 

3386.4 

3454.3 

3522.8 

10 

12 

3321.6 

3388.7 

3456.6 

3525.1 

12 

14 

3323.8 

3390.9 

3458.8 

3527.4 

14 

16 

3326.0 

3393.2 

3461.1 

3529.7 

16 

18 

3328.3 

3395.4 

3463.4 

3532.0 

18 

20 

3330.5 

3397.7 

3465.7 

3534.3 

20 

22 

3332.7 

3399.9 

3467.9 

3536.6 

22 

24 

3334.9 

3402.2 

3470.2 

3538.9 

24 

26 

3337.2 

3404.4 

3472.5 

3541.2 

26 

28 

3339.4 

3406.7 

3474.7 

3543.5 

28 

30 

3341.6 

3408.9 

3477.0 

3545.8 

30 

32 

3343  .  9 

3411.2 

3479.3 

3548.1 

32 

34 

3346.1 

3413.5 

3481.6 

3550.4 

34 

36 

3348.3 

3415.7 

3483  .  9 

3552.7 

36 

38 

3350.6 

3418.0 

3486.2 

3555.0 

38 

40 

3352.8 

3420.3 

3488.5 

3557.3 

40 

42 

3355.0 

3422.5 

3490.7 

3559.6 

42 

44 

3357.3 

3424.8 

3493.0 

3562.0 

44 

46 

3359.5 

3427.1 

3495.3 

3564.3 

46 

48 

3361.8 

3429.3 

3497.6 

3566.6 

48 

50 

3364.0 

3431.6 

3499.9 

3568.9 

50 

52 

3366.2 

3433.9 

3502.2 

3571.2  52 

54 

3368.5 

3436.1 

3504.5 

3573.5  54 

56 

3370.7 

3438.4 

3506.8 

3575.8  !  56 

58 

3373.0 

3440.7 

3509.0 

3578.1  58 

60 

3375.2 

3442.9 

3511.3 

3580.4  60 

47 


/ 

64° 

65° 

66° 

67° 

/ 

0 

3580.  4  |  3650.  4 

3721.1 

3792.6 

0 

2 

3582.8 

3652.8 

3723.4 

3795.0 

2 

4 

3585.1 

3655.1 

3725.8 

3797.4 

4 

6 

3587.4 

3657.5 

3728.2 

3799.8 

6 

8 

3589.7 

3659.8 

3730.6 

3802.2 

8 

10 

3592.1 

3662.2 

3732.9 

3804.6 

10 

12 

3594.4 

3664.5 

3735.3 

3807.0 

12 

14 

3596.7 

3666.9 

3737.7 

3809.4 

14 

16 

3599.1 

3669.2 

3740.1 

3811.8 

16 

18 

3601.4 

3671.6 

3742.4 

3814.2 

18 

20 

3603.7 

3673.9 

3744.8 

3816.  G 

20 

22 

3606.0 

3676.2 

3747.2 

3819.0 

22 

24 

3608.4 

3678.6 

3749.6 

3821.4 

24 

26 

3610.7 

3680.9 

3751.9 

3823.8 

26 

28 

3613.0 

3683.3 

3754.3 

3826.2 

28 

30 

3615.3 

3685.6 

3756.7 

3828.6 

30 

32 

3617.7 

3688.0 

3759.1 

3831.  C 

32 

34 

3620.0 

3690.4 

3761.5 

3833.4 

34 

36 

3622.3 

3692.7 

3763.9 

3835.9 

36 

38 

3624.7 

3695.1 

3766.3 

3838.3 

38 

40 

3627.0 

3697.4 

3768.7 

3840.7 

40 

42 

3629.4 

3699.8 

3771.0 

3843.1 

42 

44 

3631.7 

3702.2 

3773.4 

3845.5 

44 

46 

3634.0 

3704.5 

3775.8 

3847.9 

46 

48 

3636.4 

3706.9 

3778.2 

3850.4 

48 

50 

3638.7 

3709.3 

3780.6 

3852.8 

50 

52 

3641.1 

3711.6 

3783.0 

3855.2 

52 

54 

3643.4 

3714.0 

3785.4 

3857.6 

54 

56 

3645.7 

3716.3 

3787.8 

3860.0 

56 

58 

3648.1 

3718.7 

3790.2 

3862.5 

58 

60 

3650.4 

3721.1 

3792.6 

3864.9 

60 

48 


/ 

08° 

69° 

70° 

71° 

/ 

0 

3864.9 

3938.1 

4012.1 

4087.1 

0 

2 

3867.3 

3940.6 

4014.6 

4089.7 

2 

4 

3869.7 

3943.0 

4017.1 

4092.2 

4 

6 

3872.  % 

3945.5 

4019.6 

4094.7 

6 

8 

3874.6 

3947.9 

4022.1 

4097.2 

8 

10 

3877.0 

3950.4 

4024.6 

4099.8 

10 

12 

3879.5 

3952.9 

4027.1 

4102.3 

12 

14 

3881.9 

3955.3 

4029.6 

4104.8 

14 

16 

3884.3 

3957.8 

4032.1 

4107.3 

16 

18 

3886.8 

3960.2 

4034.6 

4109.8 

18 

20 

3889.2 

3962.7 

4037.1 

4112.4 

20 

22 

3891.6 

3965.2 

4039.6 

4114.9 

22 

24 

3894.1 

3967.6 

4042.1 

4117.4 

24 

26 

3896.5 

3970.1 

4044.6 

4119.9 

26 

28 

3898.9 

3972.5 

4047.1 

4122.4 

28 

30 

3901.4 

3975.0 

4049.6 

4125.0 

30 

32 

3903.8 

3977.5 

4052.1 

4127.5 

32 

34 

3906.3 

3980.0 

4054.6 

4130.0 

34 

36 

3908.7 

3982.4 

4057.1 

4132.6 

36 

38 

3911.2 

3984.9 

4059.6 

4135.1 

38 

40 

3913.6 

3987.4 

4062.1 

4137.7 

40 

42 

3916.1 

3989.9 

4064.6 

4140.2 

42 

44 

3918.5 

3992.3 

4067.1 

4142.7 

44 

46 

3921.0 

3994.8 

4069.6 

4145.3 

46 

48 

3923.4 

3997.3 

4072.1 

4147.8 

48 

50 

3925.9 

3999.8 

4074.6 

4150.4 

50 

52 

3928.3 

4002.2 

4077.1 

4152.9 

52 

54 

3930.8 

4004.7 

4079.6 

4155.4 

54 

56 

3933.2 

4007.2 

4082.1 

4158.0 

56 

58 

3935.7 

4009.7 

4084.6 

4160.5 

58 

60 

3938.1 

4012.1 

4087.1 

4163.1 

60 

49 


72° 

73° 

74° 

75°   ' 

4163.1 

4240.0 

4317.8 

4396.7 

0 

4165.6 

4242  .  6 

4320.5 

4399.4 

2 

4168.2 

4245.1 

4323.1 

4402.1 

4 

4170.7. 

4247.7 

4325.7 

4404.7 

6 

4173.3 

4250.3 

4328.3 

4407.4 

8 

4175.8 

4252.9 

4330.9 

4410.0 

10 

4178.4 

4255.5 

4333.6 

4412.7 

12 

4181.0 

4258.1 

4336.2 

4415.3 

14 

4183.5 

4260.7 

4338.8 

4418.0 

16 

4186.1 

4263.2 

4341.4 

4420.7 

18 

4188.6 

4265.8 

4344.0 

4423.3 

20 

4191.2 

4268.4 

4346.7 

4426.0 

22 

4193.7 

4271.0 

4349.3 

4428.6 

24 

4196.3 

4273.6 

4351.9 

4431.3 

26 

4198.8 

4276.2 

4354.5 

4434.0 

28 

4201.4 

4278.8 

4357.1 

4436.6 

30 

4204.0 

4281.4 

4359.8 

4439.3 

32 

4206.5 

4284.0 

4362.4 

4442.0 

34 

4209.1 

4286.6 

4365.1 

4444.6 

36 

4211.7 

4289.2 

4367.7 

4447.3 

38 

4214.3 

4291.8 

4370.3 

4450.0 

40 

4216.8 

4294.4 

4373.0 

4452.7 

42 

4219.4 

4297.0 

4375.6 

4455.3 

44 

4222.0 

4299.6 

4378.3 

4458.0 

46 

4224.5 

4302.2 

4380.9 

4460.7 

48 

4227.1 

4304.8 

4383.5 

4463.4 

50 

4229.7 

4307.4 

4386.2 

4466.0 

52 

4232.3 

4310.0 

4388.8 

4468.7 

54 

4234.8 

4312.6 

4391.5 

4471.4 

56 

4237.4 

4315.2 

4394.1 

4474.1 

58 

4240.0 

431Y.8 

4396.7 

4476  .  7 

60 

50 


t 

76° 

77° 

78° 

79° 

/ 

0 

4476.7 

4557.8 

4640.0 

4723.4 

0 

2 

4479.4 

4560.5 

4642.8 

4726.2 

2 

4 

4482.1 

4563.3 

4645  .  6 

4729.0 

4 

6 

4484.8 

4566.0 

4648.3 

4731.8 

6 

8 

4487.5 

4568.7 

4651.1 

4734.7 

8 

10 

4490.2 

4571.5 

4653.9 

4737.5 

10 

12 

4492.9 

4574.2 

4656.7 

4740.3 

12 

14 

4495  .  6 

4576.9 

4659.4 

4743.1 

14 

16 

4498.3 

4579.7 

4662.2 

4745.9 

16 

18 

4501.0 

4582.4 

4665.0 

4748.7 

18 

20 

4503.7 

4585.1 

4667.7 

4751.5 

20 

22 

4506.3 

4587.9 

4670.5 

4754.3 

22 

24 

4509.0 

4590.6 

4673.3 

4757.1 

24 

26 

4511.7 

4593.3 

4676.0 

4760.0 

26 

28 

4514.4 

4596.0 

4678.8 

4762.8 

28 

30 

4517.1 

4598.8 

4681.6 

4765.6 

30 

32 

4519.8 

4601.5 

4684.4 

4768.4 

32 

34 

4522.5 

4604.3 

4687.2 

4771.2 

34 

36 

4525.3 

4607.0 

4689.9 

4774.1 

36 

38 

4528.0 

4609.8 

4692.7 

4776.9 

38 

40 

4530.7 

4612.5 

4695.5 

4779.7 

40 

42 

4533.4 

4615.3 

4698.3 

4782.6 

42 

44 

4536.1 

4618.0 

4701.1 

4785.4 

44 

46 

4538.8 

4620.8 

4703.9 

4788.2 

46 

48 

4541.5 

4623.5 

4706.7 

4791.0 

48 

50 

4544.2 

4626.3 

4709.5 

4793.9 

50 

52 

4547.0 

4629.0 

4712.2 

4796.7 

52 

54 

4549  .  7 

4631.8 

4715.0 

4799.5 

54 

56 

4552.4 

4634.5 

4717.8 

4802.4 

56 

58 

4555.1 

4637.3 

4720.6 

4805.2 

58 

60 

4557.8 

4640.0 

4723.4 

4808.0 

60 

51 


' 

80° 

81° 

82° 

83°   ' 

0 

4808.0 

4893  .  9 

4981.0 

5069.4 

0 

2 

4810.9 

4896.8 

4983.9 

5072.4 

2 

4 

4813.7 

4899.7 

4986.8 

5075.4 

4 

6 

4816.6 

4902.6 

4989.8 

5078.4 

6 

8 

4819.4 

4905.4 

4992.7 

5081.4 

8 

10 

4822.3 

4908.3 

4995.7 

5084.4 

10 

12 

4825.1 

4911.2 

4998.6 

5087.3 

12 

14 

4828.0 

4914.1 

5001.5 

5090.3 

14 

16 

4830.8 

4917.0 

5004.5 

5093.3 

16 

18 

4833  .  7 

4919.9 

5007.4 

5096.3 

18 

20 

4836.5 

4922.8 

5010.3 

5099.3 

20 

22 

4839.4 

4925.7 

5013.3 

5102.3 

22 

24= 

4842.2 

4928.6 

5016.2 

5105.2 

24 

26 

48-15.1 

4931.5 

5019.2 

5108.2 

26 

2S 

4847.9 

4934.4 

5022.1 

5111.2 

28 

30 

4850.8 

4937.2 

5025.0 

5114.2 

30 

32 

4853  .  7 

4940.2 

5028.0 

5117.2 

32 

34 

4856.5 

4943.1 

5031.0 

5120.2 

34 

36 

4859.4 

4946.0 

5033.9 

5123.2 

36 

38 

4862.3 

4948.9 

5036.9 

5126.2 

38 

40 

4865.1 

4951.8 

5039.8 

5129.2 

40 

42 

4868.0 

4954.7 

5042.8 

5132.2 

42 

44 

4870.9 

4957.6 

5045.8 

5135.2 

44 

46 

4873.8 

4960.6 

5048.7 

5138.2 

46 

48 

4876.6 

4963.5 

5051.7 

5141.2 

48 

50 

4879.5 

4966.4 

5054.6 

5144.3 

50 

52 

4882.4 

4969.3 

5057.6 

5147.3 

52 

54 

4885.3 

4972.2 

5060.6 

5150.3 

54 

56 

4888.1 

4975.1 

5063.5 

5153.3 

56 

58 

4891.0 

49^8.0 

5066.5 

5156.3 

58 

60 

4893.9 

4981.0 

5069.4 

5159.3 

60 

/ 

84° 

85° 

86° 

87° 

' 

0 

5159.3 

5250.6 

5343.3 

5437.5 

0 

2 

5162.3 

5253.6 

5346.4 

5440.7 

2 

4 

5165.3 

5256.7 

5349.5 

5443  .  9 

4 

6 

5168.4 

5259.8 

5352.7 

5447.1 

6 

8 

5171.4 

5262.9 

5355.8 

5450.3 

8 

10 

5174.4 

5266.0 

5358.9 

5453.4 

10 

12 

51-77.5 

5269.0 

5362.0 

5456.6 

12 

14 

5180.5 

5272.1 

5365.2 

5459.8 

14 

16 

5183.5 

5275.2 

5368.3 

5463.0 

16 

18 

5186.6 

5278.3 

5371.4 

5466.2 

18 

20 

5189.6 

5281.4 

5374.6 

5469.4 

20 

22 

5192.6 

5284.4 

5377.7 

5472.5 

22 

24 

5195.6 

5287.5 

5380.8 

5475  .  7 

24 

26 

5198.7 

5290.6 

5383.9 

5478.9 

26 

28 

5201.7 

5293.7 

5387.1 

5482.1 

28  - 

30 

5204.7 

5296.7 

5390.2 

5485.3 

30 

32 

5207.8 

5299.8 

5393.4 

5488.5 

32 

34 

5210.8 

5302.9 

5396.5 

5491.7 

34 

36 

5213.9 

5306.1 

5399.7 

5494.9 

36 

38 

5216.9 

5309.2 

5402.8 

5498  .  1 

38 

40 

5220.0 

5312.3 

5406.0 

5501.3 

40 

42 

5223.1 

5315.4 

5409.1 

5504.5 

42 

44 

5226.1 

5318.5 

5412.3 

5507.7 

44 

46 

5229.2 

5321.6 

5415.4 

5510.9 

46 

48 

5232.2 

5324.7 

5418.6 

5514.1 

48 

50 

5235.3 

5327.8 

5421.8 

5517.3 

50 

52 

5238.3 

5330.9 

5424.9 

5520.5 

52 

54 

5241.4 

5334.0 

5428.1 

5523.7 

54 

56 

5244.5 

5337.1 

5431.2 

5526.9 

56 

58 

5247.5 

5340.2 

5434.4 

5530.1 

58 

60 

5250.6 

5343.3 

5437.5 

5533.3 

60 

53 


/ 

88° 

/ 

88° 

/ 

se° 

/ 

89° 

0 

5533.3 

30 

5581.9 

0 

5630.8 

30 

5680.2 

2 

5536.6 

32 

5585.1 

2 

5634.1 

32 

5683.5 

4 

5539.8 

34 

5588.4 

4 

5637.4 

34 

5686.8 

6 

5543.1 

36 

5591.7 

6 

564C.7 

36 

5690.2 

8 

5546.3 

38 

5594.9 

8 

5644.0 

38 

5693.5 

10 

5549.5 

40 

5598.2 

10 

5647.3 

40 

5696.8 

12 

5552.8 

42 

5601.4 

12 

5650.6 

42 

5700.1 

14[5556.0 

44 

5604.7 

14 

5653.9 

44 

5703.4 

165559.2 

46 

5608.0 

16 

5657.1 

46 

5706.8 

18 

5562.5 

48 

5611.2 

18 

5660.4 

48 

5710.1 

20 

5565.7 

50 

5614.5 

20 

5663.7 

50 

5713.4 

22 

5568.9 

52 

5617.8 

22 

5667.0 

52 

5716.7 

84 

5572.2 

54 

5621.0 

24 

5670.3 

54 

5720.0 

2(\ 

5575.4 

56 

5624.3 

26 

5673.6 

56 

5723.4 

2* 

5578.6 

58 

5627.5 

28 

5676.9 

58 

5726.7 

30 

5581.9 

60 

5630.8 

30 

5680.2 

60 

5730.0 

TABLE 

GIVING  BADII  OF  DEGEEES 
OF  CURVE. 


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UNIVERSITY    OF    CALIFORNIA 
LIBRARY 


No. 


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81.- 


No.  82.- 


No. 
No. 

No. 
No. 


83.- 
84.- 

85.- 


No.  87.- 

No.  88.- 
No.  89.- 


No. 
No. 
No. 
No. 

No. 
No. 
No. 
No. 
No. 


94. 
95. 
96. 
97. 


Due  two  weeks  after  date. 
OCT   10  1913 


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No.  99.-r. 


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Kennedy.    The  Transnv  / 

Power  from  Central  S' 

By  Prof.  W.  C.  Unw' 
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A  Rational  an'*  f  A 

Stresses  in  r 

Introductir 

Statics.    " 
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No.  109 


H'l.l 


All 

.iical 

6. 

*  Method 

By  Lloyd 

instructor,  U. 


CUR- 

g  Instruments.    By 

<or  Electrical  Energy. 

Edited,  with  Preface, 

.rtin.    Folding  Plate  and 


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